Philosophy Dictionary of Arguments

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Numbers: whether numbers are objects or concepts, has been controversial in the philosophical discussion for millennia. The most widely accepted definition today is given by G. Frege (G. Frege, Grundlagen der Arithmetik 1987, p. 79ff). Frege-inspired notions represent numbers as classes of classes, or as second-level terms, or as that with one measure the size of sets. Up until today, there is an ambiguity between concept and object in the discussion of numbers. See also counting, sets, measurements, mathematics, abstract objects, mathematical entities, theoretical entities, number, platonism.

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

Author Concept Summary/Quotes Sources

Gottlob Frege on Numbers - Dictionary of Arguments

II 18 f
Numbers/Frege: e.g. 16 = 4², 4 x 4 = 4². Here we see that equality of meaning does not lead to equality of thought.
>Fregean sense, >Fregean meaning, >Thoughts, >Equality, >Equations.
II 66 ff
The figure contains the expression of a concept. >Concepts.
Properties will be expressed by a concept. A concept may fall under a higher one. E.g. there is at least one square root of 4. This is not a statement about a certain number 2, nor about -2, but about a concept, namely the square root of 4.
II 81 f
There are no variable numbers. Variable: do we not denote variable numbers by x, y, z? This way of speaking is used, but these letters are not proper names of variable numbers, like "2" and "3" are proper names of constant numbers. We cannot specify which properties "x" has in contrast to y.
Variable: is not a proper name of an indefinite or variable number. X has no properties (only in the context). "Indefinitely" is not an adjective, but an adverb for the process of calculating.
Generality/Frege: generality is not a meaning but a hint.
Proper Names: π, i, e are not variables!
Generality: here, the number has to play two roles: as an object it is called a variable, as a property, it is called a value.
Function: has generality, is a law. To any number of the x-range a number from the y-range is assigned. A function is not a variable! (An elliptic function is not an elliptic variable). The function is unsaturated. >Unsaturated.
II 77
Number/object/calculating/addition/Frege: only from the meaning of the words "the number 4" (Frege: = object) we can say that it is the result of combining 3 and 1. Not of the concept. Calculation result: is an object, the result of the calculation: is not a concept.
II 85
Number/Frege: e.g. "a variable takes on a value". Here, the number has to play two roles: as an object it is called a variable, as a property, it is called a value.
- - -
I 38
Numbers/Frege: from physical observations no conclusions can be drawn about numbers.
I 47
Quantity/Frege: quantity is a concept. Number: is an object. >Objects.
I 48
Numbers/Newton: numbers are the ratio of each size to another. FregeVsNewton: here, the notions of size and ratio are presupposed.
I 49
Numbers/Frege: Problem: numbers as sets: here, the concept of quantity is pressupposed.
I 60
Number/Frege: number is no multiplicity. That would exclude 0 and 1.
I 62
Number/one/unit/property/Frege: "One" cannot be a property. Otherwise, there would be no thing that does not have this property.
I 82
Not the objects but the concepts are the bearers of the number. Otherwise, different numbers could be assigned to the same example. Thus the abstraction is accompanied by a judgment.
I 90
A number is not the property of a concept. Number: is an abstract object, not a property -> see below. Number Equality/equality: number equality is a concept (not an object).
I 100/101
Def Quantity/Frege: the quantity which belongs to the concept F is the scope of the concept equal numbered to the concept F.
I 100
Scope/concept scope/Frege: if the straight a is parallel to straight b, then the scope of the concept of straight parallel to straight a is equal to the scope of the concept straight parallel to the straight b and vice versa - scope equality. >Term scope, >Equality.
I 110
Number/Frege/(s): comes from the distinction concept term scope (quantity)/object (number). If the object is zero, the quantity that belongs to this concept is one.
((s) This is how Frege gets from 0 to 1: one is the number-of objects falling under the concept "equal-to-zero", namely one object. Zero ist the number of objects falling under the concept "equal-to-zero-and-not-equal-to-zero").
>Zero, >One.
I 121
Numbers/Frege: numbers are not concepts. They are (abstract) objects (see above). Quantities are concepts.
I 128
Term: e.g. square root of -1. This cannot be used with the definite article.
I 135
Number/Frege: a number is neither heaps of things, nor a property of such.
I 130
Number system/expansion/Frege: in the expansion, the meaning is not be established arbitrarily. E.g. the meaning of the square root is not already invariably established before the definitions, but it is determined by them. ((s) Frege: wants to point at the meaning as use within a system.). The new numbers are given to us as scopes of concepts.
I 136
Each figure is an equation. >Equations.
- - -
Berka I 83
Number/Frege: numbers must be defined in order to be able to present completeness of evidence at all - (> sequence).(1)

1- G. Frege, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle 1879, Neudruck in: Ders. Begriffsschrift und andere Aufsätze, hrsg. v. J. Agnelli, Hildesheim 1964

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments
The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

G. Frege
Die Grundlagen der Arithmetik Stuttgart 1987

G. Frege
Funktion, Begriff, Bedeutung Göttingen 1994

G. Frege
Logische Untersuchungen Göttingen 1993

Berka I
Karel Berka
Lothar Kreiser
Logik Texte Berlin 1983

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Ed. Martin Schulz, access date 2023-06-03
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